.
The formula for the curved surface area of a cone which does not include the area of the base is:
A=pirl where r is the radius of the base and l is the lateral height (slant height) as shown in yellow below:
![enter image source here]()
From the right angle triangle shown, using Pythagoras' formule, we get:
l^2=h^2+r^2
l=sqrt(h^2+r^2
Therefore, the curved area A is:
A=pirsqrt(h^2+r^2)
The volume formula is:
V=1/3pir^2h
Let's solve for h from the area formula and substitute it into the volume formula:
A^2=pi^2r^2(h^2+r^2)
A^2=pi^2r^2h^2+pi^2r^4
pi^2r^2h^2=A^2-pi^2r^4
h^2=(A^2-pi^2r^4)/(pi^2r^2)
h=sqrt(A^2-pi^2r^4)/(pir)
V=1/3pir^2sqrt(A^2-pi^2r^4)/(pir)
V=1/3rsqrt(A^2-pi^2r^4)=1/3r(A^2-pi^2r^4)^(1/2)
To find the maximum volume, we take the derivative of the volume function and set it equal to 0:
(dV)/(dr)=1/3[r(1/2)(A^2-pi^2r^4)^(-1/2)(-4pi^2r^3)+(A^2-pi^2r^4)^(1/2)]
(dV)/(dr)=1/3((-2pi^2r^4)/sqrt(A^2-pi^2r^4)+sqrt(A^2-pi^2r^4))
(dV)/(dr)=(-2pi^2r^4+A^2-pi^2r^4)/(3sqrt(A^2-pi^2r^4)
(dV)/(dr)=(-3pi^2r^4+A^2)/(3sqrt(A^2-pi^2r^4))=0
-3pi^2r^4+A^2=0
A^2=3pi^2r^4
Let's substitute for A from the area formula:
pi^2r^2(h^2+r^2)=3pi^2r^4
Dividing both sides by pi^2r^2, we get:
h^2+r^2=3r^2
h^2=2r^2
h^2/r^2=2
h/r=sqrt2/1
